Beyond the GW approximation - APS Link Manager - American ...

BEYOND THE GW APPROXIMATION: COMBINING ... PHYSICAL REVIEW B 85, 155131 (2012)
the T matrix reads,
Ō σ 1 ¯σ 1 ,(2)
ij
(ω) = Ō σ 1 ¯σ 1 ,(1)
ij
(ω) + U ∑ 3 n
and the two self-energies become
pp,σ 1,(2)
ij
(ω)
= σ 1,(1)
ij
(ω) + U 3 {
16
[
− 1 2t
1
ω − 3t + iη +
eh,σ 1 (2)
ij
(ω)
= σ 1,(1)
ij
(ω) − U 3
16
− 1 2t
[
{
1
ω − 3t + iη +
L σ 1 ¯σ 1
0,in (ω)Lσ 1 ¯σ 1
0,nj
(ω), (59)
1
(ω − 3t + iη) − (−1)(i−j)
2 (ω + 3t − iη) 2
(−1)(i−j)
ω + 3t − iη
]}
, (60)
1
(ω − 3t + iη) − (−1)(i−j)
2 (ω + 3t − iη) 2
(−1)(i−j)
ω + 3t − iη
]}
. (61)
Combining the two interaction channels by adding Eqs. (60)
and (61), the second terms on the right-hand side of the two
equations cancel each other, thus restoring the exact result if
1
2 (pp,(2) + eh,(2) ) is taken. We have already shown in Eq. (25)
that the sum 1 2 (pp,(2) + eh,(2) ) takes into account some of the
terms that would appear in the self-energy if also the functional
derivative δO/δG were considered. This might justify the
exact result that is obtained by taking 1 2 (pp,(2) + eh,(2) ). Also
the sum ( pp,(2) + eh,(2) ) takes into account some of the terms
arising from the functional derivative [see Eq. (25)]; however,
it does not give the exact result. In other words it seems more
important to take into account the term −T 0 than (T 1 + T 2 )/2
[see Eqs. (24) and (25)] at least in the present problem.
However, with the third iteration, the fourth-order terms in
the pp and eh T -matrix self-energies are the same and they
would not cancel each other if the sum 1 2 (pp,(2) + eh,(2) )is
taken. Instead, with the fourth iteration, the fifth-order terms in
the pp and eh T -matrix self-energies would cancel each other.
In general, for the present problem, the pp and eh T -matrix
self-energies show, starting from second order, the same
even-order terms and opposite odd-order terms, as one can
verify Taylor expanding the frequency-dependent part of the
pp and eh self-energies [Eqs. (53) and (54)] for small U. The
same holds for the Hubbard model at 1/4 filling. Therefore,
summing the two contributions will not give the exact result.
Even adding the GW self-energy terms and its exchange
counterparts, in the spirit of the FLEX approximation, will
not produce the exact result. These findings show that there
is no an ultimate way to combine diagrams, and this is of
clear relevance for realistic studies where several attempts
to combine pp and eh channels have been done (see, e.g.,
Refs. 28–30 and 35). A possibility is to use the screened T
matrix we introduced in Sec. II B3, which produces, at least
for the studied problem, results that are overall better than
those from the GW and T matrix, as shown in the following.
2. Screened T matrix
The pp screened T matrix with the approximate on-site
static Coulomb interaction W = U − 2U 2 t/h 2 (with h 2 =
4t 2 + 4Ut) leads to the self-energy
pp,σ
s,ij (ω) = U 2 δ ij + UWt [
1
4 ˜h ω − t − ˜h + iη
]
+ (−1)(i−j)
ω + t + ˜h − iη
(62)
10
6000
4000
exact and 1st iteration pp/eh T matrix
pp T matrix
pp screened T matrix
GW
U/2t
5
2000
ω/t
0
∝ √U/2t
-2000
-5
-4000
-U/2t
-10
0
2
4
U/t
6
8
-6000
10
2 3 4 5 6 7 8 9
10 4
Log (U/t)
t . A σ (ω)
FIG. 7. Two-site Hubbard model at 1/2 filling: Comparison between the exact renormalized addition/removal energies ω/t (solid lines) as
a function of U/t (left panel) and log(U/t) (right panel) and the results obtained from GW (dashes) and particle-particle (dots) and screened
T matrix (triangles). In the atomic limit, the spectral function, i.e., the peak positions and weights, is illustrated on the right-hand side, on
multiplying by t and taking the t → 0 limit.
155131-11